A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

I have the answer but i dont know how.. The lower edge of a mural, 12 ft high, is 6ft above an observer’s eye. Under the assumption that the most favorable view is obtained when the angle subtended by the mural at the eye is maximum, at what distance from the wall should the observer stand?

  • This Question is Closed
  1. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433509816192:dw|

  2. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    we have the same drawing sir

  3. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    From the lower small triangle \[\tan(s) = \frac{6}{x}\tag{1}\] From the big triangle \[\tan(t+s)=\frac{18}{x}\tag{2}\]

  4. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Our goal is to eliminate \(s\) somehow and get an equation with \(t\) and \(x\) as variables

  5. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    In view of that, apply the angle sum formula : \[\large \tan(t)=\tan((t+s)-s)=\frac{\tan(t+s)-\tan(s)}{1+\tan(t+s)\tan(s)}\] plugin the values

  6. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok sir i will solve now

  7. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i got this sir: |dw:1433510419375:dw|

  8. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    doesn't look correct

  9. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\large{\begin{align} \tan(t)=\tan((t+s)-s) &=\frac{\tan(t+s)-\tan(s)}{1+\tan(t+s)\tan(s)}\\~\\ &=\dfrac{\dfrac{18}{x}-\dfrac{6}{x}}{1+\dfrac{18}{x}\dfrac{6}{x}}\\~\\ &=\dfrac{12x}{x^2+108} \end{align}}\]

  10. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so the equation is \[\large \tan(t)=\dfrac{12x}{x^2+108}\] and we want to maximize \(t\)

  11. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i got that equation sir

  12. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    may be isolate \(t\) and find \(\dfrac{dt}{dx}\)

  13. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is this the bases sir? |dw:1433510949377:dw|

  14. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\large \tan(t)=\dfrac{12x}{x^2+108}\] \[\large t = \tan^{-1}\left(\dfrac{12x}{x^2+108}\right)\] \[\large \dfrac{dt}{dx} = ?\]

  15. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    remember the derivative of \(\tan^{-1}(x)\) ?

  16. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433511150272:dw|

  17. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes use that

  18. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\large t = \tan^{-1}\left(\dfrac{12x}{x^2+108}\right)\] \[\large{\begin{align} \dfrac{dt}{dx} &= \dfrac{1}{1+\left(\dfrac{12x}{x^2+108}\right)^2}*\dfrac{d}{dx}\left(\dfrac{12x}{x^2+108}\right)\\~\\~\\ &=? \end{align}}\]

  19. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok wait sir i will solve

  20. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i got this sir:

  21. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\frac{ 1 }{ 1+\frac{ 12X ^{2} }{ x ^{4}+216x ^{2}+11664 } }\times \frac{ -12x ^{2}+1296 }{ x ^{4}+216x ^{2}+11664 }\]

  22. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ohh there an error im sorry.. it should be 144x^2 instead of having 12x^2

  23. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and i was stuck into this.. i dont know how to continue...

  24. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i already have an headache..

  25. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    do not expand

  26. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    because we will be setting all that equal to 0 and denominator gona vanish

  27. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\large{\begin{align} \dfrac{dt}{dx} &= \dfrac{1}{1+\left(\dfrac{12x}{x^2+108}\right)^2}*\dfrac{d}{dx}\left(\dfrac{12x}{x^2+108}\right)\\~\\~\\ &=\dfrac{1}{1+\left(\dfrac{12x}{x^2+108}\right)^2}*\left(\dfrac{(x^2+108)12-12x(2x)}{(x^2+108)^2}\right)\\~\\~\\ \end{align}}\] yes ?

  28. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    setting that equal to 0 gives \[\large (x^2+108)12-12x(2x) = 0\] solve \(x\)

  29. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    x=10.4

  30. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    but sir im a little bit confused by this... how come the remaining equation becomes \[(x ^{2}+108)12-12x(2x)\]

  31. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    how did the others vanish? and this is the equation that still remains?

  32. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    where did this \[\frac{ 1 }{ 1 +\left( \frac{ 12x }{ x ^{2}+108 } \right)^{2}}\] go?

  33. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.